Stochastic Quasi-1D Percolation Paths

• The paper identifies that standard finite-size scaling fails in quasi-one-dimensional percolation, highlighting nonuniversal thresholds and abrupt, explosive transitions.
• It employs hierarchical small-world modifications and eigenvalue analyses to reveal discontinuous order parameters and critical clustering effects.
• Exact coupling and regeneration techniques demonstrate ergodicity and large deviation principles, linking percolation behavior to anomalous diffusion and SLE frameworks.

Stochastic quasi-one-dimensional percolation paths describe random connectivity phenomena in systems constrained to narrow or effectively one-dimensional geometries, where stochasticity arises via randomness in connectivity, site/bond occupation, geometry, or external perturbations. These paths play a central role in a range of physical, mathematical, and applied contexts—including nanostructure transport, thin-film conductance, small-world networks, random polymers, drainage networks, and quantum communication. The theoretical investigation integrates percolation thresholds, scaling exponents, ergodic and large deviations behavior, explosive transitions, and the effect of disorder or reinforcement, uncovering features sharply distinct from higher-dimensional percolation models.

1. Percolation Thresholds and Scaling Failure in One Dimension

One-dimensional percolation theory fundamentally differs from its higher-dimensional counterparts due to the absence of a nontrivial phase transition in the classical homogeneous case: only at full bond or site density does a spanning cluster emerge. For stochastic quasi-one-dimensional systems, randomness is typically introduced via random occupation of bonds/sites, local inhomogeneities, or reinforcements.

The percolation threshold in a chain of N sites or bonds is characterized by the relation $x = N_1 / N$ for sites, and, via Hammersley’s inequality, $x_{cb} \le x$ for bonds (Bureeva et al., 2011). Importantly, finite-size effects and the underlying randomness manifest in the determination of connectivity: for percolation radius $R = 2, 3$, thresholds align with these relations, but can vary with system geometry or imposed randomness.

Critical exponents for correlation length ($\nu$) and specific heat ($\alpha$) demonstrate anomalously large values, especially for bond-type randomness, diverging steeply near the threshold. The scaling-like relation $\sigma(N) \propto N^{-\nu}$ yields $\nu$ significantly exceeding the mean-field expectation ($\nu = 1/2$), indicating hypersensitivity in spatial convergence and cluster growth. The system’s free energy shows scaling $F(x) \propto |x - x_c|^{2 - \alpha}$. Crucially, the stability condition $\alpha + \nu d \geq 2$ is always strictly satisfied, with $\alpha + \nu \approx 4.8$ in the infinite system limit ($d=1$), violating the classical scaling degeneracy $\alpha + \nu d = 2$ (Bureeva et al., 2011).

This violation implies standard finite-size scaling methods are inapplicable in quasi-one-dimensional stochastic systems; the percolation transition is highly non-universal and dominated by system-specific corrections.

2. Explosive Percolation Transitions in Small-World and Hierarchical Geometries

Introduction of hierarchical or small-world bonding alters the percolation landscape. In one-dimensional lattices dressed with recursive long-range bonds, system behavior shifts from “trivial” (percolation only at $p=1$) to exhibiting an explosive, discontinuous transition at $p_c = 1/2$ (Boettcher et al., 2011). The order parameter, $P_\infty(p)$, jumps discontinuously at $p_c$ from zero to a finite value (e.g., $P_\infty(p_c) \approx 0.60979\ldots$), with the recursive connectivity probability $T_{n+1} = p + (1-p)T_n^2$ spawning subextensive clusters below $p_c$ and extensive ones above.

Cluster size scaling is governed by the largest eigenvalue $\lambda$ of the generating function recursion, with $\Psi(p) = (\ln \lambda)/(\ln 2)$ controlling the fractality: for $p < 1/2$, clusters are fractal; at $p = 1/2$ a sudden switch to extensive clustering occurs. These findings indicate the critical importance of geometric and topological construction in enabling abrupt connectivity transitions in quasi-one-dimensional stochastic models.

Hierarchically induced transitions highlight how “small-world” link additions can fundamentally alter the threshold and critical behavior of otherwise subcritical or trivial percolation systems; such mechanisms may underpin rapid epidemic spread, information transport in narrow-band networks, or cascading failures.

3. Regeneration, Coupling, and Ergodicity in Stochastic Paths

Regeneration properties and percolation-induced ergodicity emerge as foundational in stochastic quasi-one-dimensional dynamics. In voter models with local copying, boundary nucleation, and bulk randomization, genealogies are encoded via coalescing, branching, and dying random walks (Mohylevskyy et al., 2011). Bulk nucleation events truncate genealogies, while boundary nucleation exposes branching structure when neighbors disagree; ergodicity occurs when all genealogical paths terminate almost surely—quantified by the absence of infinite percolation paths in the “arrow graph.”

Enhancement arguments extend ergodicity even in parameter regimes supporting infinite genealogies, by showing discrepancies between coupled color histories eventually dissipate. Thus, ergodicity and convergence to a unique invariant measure are ensured provided percolation-based criteria for finite genealogies or renormalized discrepancy extinction are satisfied.

In first-passage percolation models, regenerative structures allow for coupling of processes and derivation of 0-1 laws for tail σ-algebras (Ahlberg, 2011). Exact coupling ensures post-coupling independence from initial configurations, yielding trivial tail σ-algebras: any event depending on the infection after all finite times takes probability either zero or one.

Regeneration and exact coupling are critical for the rigorous probabilistic description of long-term behavior (ergodicity, growth rates, 0-1 laws) in stochastic quasi-one-dimensional percolation paths.

4. Directional Growth, Subadditivity, and Large Deviations

Stochastic quasi-one-dimensional paths in supercritical oriented percolation feature exponential proliferation, with the number of open paths of length $n$, $N_n$, showing almost-sure convergence of $N_n^{1/n}$ to a deterministic constant under the survival event (Garet et al., 2013). The limit log-growth rate $\widetilde{\alpha}_p(0)$ is explicitly established via subadditive ergodic theory, regeneration times, and trajectory adaptation.

Limiting directional behavior is encoded by a concave, continuous rate function $\widetilde{\alpha}_p(x)$ on the unit ball of an appropriate norm: for paths ending near direction $x$, $$(1/n)\log N_{nA,n} \to \sup_{x \in A} \widetilde{\alpha}_p(x)$$. Key ingredients include essential hitting times and renewal structure, with adapted sequences $S_n(y,h)$ ensuring subadditive concatenation and regeneration along quasi-one-dimensional directions.

These results assert both exponential large deviations principles for endpoints of uniformly chosen open paths and a connection to random polymer models, highlighting sharp growth and concentration in stochastic quasi-one-dimensional regimes.

5. Domination, Lipschitz Embedding, and Obstacle Reduction

In systems where random obstacles or defects block quasi-one-dimensional paths, stochastic domination methods show that the union of overlapping intervals (e.g., obstacles modeled as random sticks or balls) is probabilistically dominated by an i.i.d. percolation process with controlled parameter (Holroyd et al., 2012). The “balls and sticks” principle—projecting higher-dimensional obstacles onto one-dimensional overlaps—facilitates reduction of complex dependent environments to one-dimensional analogs.

Lipschitz embedding techniques demonstrate the existence (and construction) of infinite paths or comb graphs in percolation models, provided obstacles are sparse or have exponentially decaying tails. When moments of random radii are finite, the corresponding reinforced region remains subcritical, and percolation fails even along the reinforced path (Nascimento et al., 12 Jun 2024). The existence of an embedding (curves with bounded slope avoiding obstacles) directly answers questions regarding the feasibility of long, well-behaved quasi-one-dimensional paths in disordered media.

Such domination and embedding approaches clarify how local randomness, defect constraints, and reinforcement impact global connectivity, allowing precise quantification of percolation thresholds and existence theorems for quasi-one-dimensional structures in stochastic environments.

6. Extreme Path Ordering, Monotonicity, and Correlations

In oriented percolation models, the probability distributions of leftmost or rightmost open paths exhibit natural monotonicity and stochastic ordering properties (Andjel et al., 2014). Adding points to starting or ending sets on one side increases the probability that the extreme path shifts in the same direction. These features extend to related models, such as the discrete-time contact process and site percolation, due to the preservation of extreme path uniqueness and invariance under local updates.

A direct consequence is a counterintuitive negative correlation between leftmost and rightmost path events conditioned on an intermediate event: conditioning on connection to a “middle” site increases the likelihood of both extreme events, a phenomenon deriving from the geometric partial ordering. These findings give precise control over the spatial distribution and statistical dependencies among extremal paths in two-dimensional or quasi-one-dimensional percolation geometries.

7. Anomalous Diffusion and SLE in Anisotropic Quasi-One-Dimensional Paths

Anisotropic percolation paths, particularly those in multi-layered or directed models, are naturally described via stochastic Loewner evolution (SLE) with driving functions that display anomalous diffusion—i.e., long-range correlated, non-Markovian stochastic processes (Credidio et al., 2015). The mean square displacement of the driving function, $\langle U_t^2 \rangle \sim t^\alpha$, deviates from the Brownian value $\alpha=1$; superdiffusive ($\alpha \approx 1.78$) or subdiffusive ($\alpha \approx 0.67$) behavior captures directional or layered bias in percolation cluster growth.

Numerical extraction of driving functions and simulations using fractional Brownian motion (with variable Hurst exponent) corroborate the link between anomalous stochastic process properties and fractal anisotropy in percolation perimeter scaling. These generalized SLE representations provide a mathematically rigorous framework for interpreting non-Markovian effects at criticality and establishing connections between anomalous diffusion and geometric roughness in quasi-one-dimensional percolation interfaces.

8. Shielded Paths, Complement Percolation, and Multi-Phase Transitions

In independent bond percolation on $\mathbb{Z}^d$, shielded paths—self-avoiding routes where every vertex has all incident edges closed—yield infinite structures entirely contained in the complement of the infinite open cluster (Bock et al., 2018). These paths only exist above a critical “shielded” threshold $p_{\textrm{shield}}(d)$, with both rigorous upper bounds (involving self-avoiding walk connective constants) and lower bounds (via second-moment arguments), showing $p_{\textrm{shield}} > p_c$ for $d \geq 10$ (possibly $d \geq 7$ numerically).

Such shielded structures characterize the geometry of the vacant region post-infinite cluster emergence and are connected to multi-phase transitions, where percolation in the complement persists beyond the classical threshold. This framework reveals deep interplay between local constraints (vertex shielding) and global connectivity, with direct implications for frozen percolation, robustness analysis, and phase diagram structure in high-dimensional stochastic systems.

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Stochastic quasi-one-dimensional percolation paths thus encapsulate a rich domain of probabilistic, geometric, and physical phenomena, where randomness interacts with spatial constraint to produce nontrivial connectivity, scaling anomalies, explosive transitions, regenerative ergodicity, and multiscale structure. These features inform theory and application in diverse fields, from condensed-matter transport and disordered materials to epidemic modeling, quantum networks, and landscape ecology.